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Error Estimation for Eigenvalues of Unbounded Linear Operators and an Application to Energy Levels in Graphene Quantum Dots
Abstract
The eigenvalue problem for linear differential operators is important since eigenvalues correspond to the possible energy levels of a physical system. It is also important to have good estimates of the error in the computed eigenvalues. In this work we use spline interpolation to construct approximate eigenfunctions of a linear operator by using the corresponding eigenvectors of a discretized approximation of the operator. We show that an error estimate for the approximate eigenvalues can be obtained by evaluating the residual for an approximate eigenpair. The interpolation scheme is selected in such a way that the residual can be evaluated analytically. In order to demonstrate that the method gives useful error bounds we apply it to a problem originating from the study of graphene quantum dots where the goal was to investigate the change in the spectrum from incorporating electron-electron interactions in the potential.
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We study the effect of electron-electron interaction in graphene quantum dots defined by an external electrostatic potential and a high magnetic field. To account for the electron-electron interaction, we use the Thomas-Fermi approximation and find that electron screening causes the formation of compressible strips in the potential profile and the electron density. We numerically solve the Dirac equations describing the electron dynamics in quantum dots, and we demonstrate that compressible strips lead to the appearance of plateaus in the electron energies as a function of the magnetic field. Finally, we discuss how our predictions can be observed using the Kelvin probe force microscope measurements.
Dirac fermions interacting with a cylindrically symmetric quantum dot potential created in single and bilayer graphene are not confined but form quasibound states. The broadening of these quasibound states (i.e., the inverse of their lifetimes) decreases (increases) with the orbital momentum of the electron in the case of graphene (bilayer). Quasibound states with energy below (above) the barrier height are dominantly electronlike (holelike). A remarkable decrease of the energy level broadening is predicted for electron energies close to the barrier height, which are a consequence of the total internal reflection of the electronic wave at the dot edge.
We examine a graphene quantum dot formed by combining an electric and a
uniform magnetic field. The electric field creates a smooth quantum well
potential while the magnetic field induces an exponential tail to the dot
states. The states peak in the well and the electrostatic barrier region as a
result of the Klein tunneling effect. Coupling between dot states which peak in
different regions can be achieved with the electric and magnetic fields. The
tunability of this dot with moderate external fields could be used for
designing quantum devices in monolayer graphene.
It is shown that a magnetic field dramatically affects the problem of
supercritical charge in graphene making any charge in gapless theory
supercritical. The cases of radially symmetric potential well and Coulomb
center in an homogeneous magnetic field are considered. The local density of
states and polarization charge density are calculated in the first order of
perturbation theory. It is argued that the magnetically induced instability of
the supercritical Coulomb center can be considered as a quantum mechanical
counterpart of the magnetic catalysis phenomenon in graphene.
Quantum dots induced by a strong magnetic field applied to a single layer of graphene in the perpendicular direction are investigated. The dot is defined by a model potential which consists of a well of depth relative to a flat asymptotic part and quantum states formed from the zeroth Landau level are considered. The energy of the dot states cannot be lower than relative to the asymptotic potential. Consequently, when is chosen to be about half of the gap between the zeroth and first Landau levels, the dot states are isolated energetically in the gap between Landau level 0 and Landau level -1. This is confirmed with numerical calculations of the magnetic field dependent energy spectrum and the quantum states. Remarkably, an antidot formed by reversing the sign of also confines electrons but in the energy region between Landau level 0 and Landau level +1. This unusual behaviour gives an unambiguous signal of the novel physics of graphene quantum dots. Comment: 4 pages, 2 figures
We present an analytical theory for the gate electrostatics and the classical
and quantum capacitance of the graphene nanoribbons (GNRs) and compare it with
the exact self-consistent numerical calculations based on the tight-binding
p-orbital Hamiltonian within the Hartree approximation. We demonstrate that the
analytical theory is in a good qualitative (and in some aspects quantitative)
agreement with the exact calculations. There are however some important
discrepancies. In order to understand the origin of these discrepancies we
investigate the self-consistent electronic structure and charge density
distribution in the nanoribbons and relate the above discrepancy to the
inability of the simple electrostatic model to capture the classical gate
electrostatics of the GNRs. In turn, the failure of the classical
electrostatics is traced to the quantum mechanical effects leading to the
significant modification of the self-consistent charge distribution in
comparison to the non-interacting electron description. The role of
electron-electron interaction in the electronic structure and the capacitance
of the GNRs is discussed. Our exact numerical calculations show that the
density distribution and the potential profile in the GNRs are qualitatively
different from those in conventional split-gate quantum wires; at the same
time, the electron distribution and the potential profile in the GNRs show
qualitatively similar features to those in the cleaved-edge overgrown quantum
wires. Finally, we discuss an experimental extraction of the quantum
capacitance from experimental data.
We analytically calculate the energy spectrum of a circular graphene quantum dot with radius R subjected to a perpendicular magnetic field B by applying the infinite-mass boundary condition. We can retrieve well-known limits for the cases R, B going to infinity and B going to zero. Our model is capable of capturing the essential details of recent experiments. Quantitative agreement between theory and experiment is limited due to the fact that a circular dot is not close enough to the experimental geometry, that disorder plays a significant role, and that interaction effects may be relevant.
Massless Dirac particles cannot be confined by an electrostatic potential. This is a problem for making graphene quantum dots but confinement can be achieved with a magnetic field and here, general conditions for confined and deconfined states are derived. There is a class of potentials for which the character of the state can be controlled at will. Then a confinement-deconfinement transition occurs which allows the Klein paradox to be probed experimentally in graphene dots. Comment: 4 pages, 4 figres
This paper presents an overview of parallel algorithms and their implementations for solving large sparse linear systems which arise in scientific and engineering applications. Preconditioners constitute the most important ingredient in solving such systems. As will be seen, the most common preconditioners used for sparse linear systems adapt domain decomposition concepts to the more general framework of “distributed sparse linear systems”. Variants of Schwarz procedures and Schur complement techniques are discussed. We also report on our own experience in the parallel implementation of a fairly complex simulation of solid-liquid flows.
We find that a repulsive potential of graphene in the presence of a magnetic field has bound states that are peaked inside the barrier with tails extending over l(N+1), where l and N are the magnetic length and Landau level index. We have investigated how these bound states affect scaling properties of the induced density of filled Landau levels of massless Dirac fermions. For chiral fermions, we find, in strong coupling regime, that the density inside the repulsive potential can be greater than the value in the absence of the potential, while in the weak coupling regime, we find negative induced density. Similar results also hold for nonchiral fermions. As one moves from weak to strong coupling regimes, the effective coupling constant between the potential and electrons becomes more repulsive, and then it changes sign and becomes attractive. Different power laws of induced density are found for chiral and nonchiral fermions.
The effect of a magnetic field on the charged vacuum is investigated. The
field dependence of the energy levels causes jumps in the total vacuum charge
that occur whenever an energy level crosses the Fermi level and this leads to
re-entrant cycles of vacuum charging and discharging. In atomic systems these
effects require astrophysical magnetic fields of around 10^8 T but in graphene
with a mass gap they occur in laboratory fields of about 1 T or lower. It is
suggested that an electrostatic graphene quantum dot defined by a gate
electrode provides a solid state model of the as yet unobserved charged vacuum
as well as a model of an atomic system in an extreme astrophysical environment.
Phase diagrams are computed to show how the total vacuum charge depends on the
confining potential strength and applied magnetic field. In addition the field
dependence of the vacuum charge density is investigated and experimental
consequences are discussed.
We report on several new basic properties of a parabolic dot in the presence of a magnetic field. The ratio between the potential strength and the Landau level (LL) energy spacing serves as the coupling constant of this problem. In the weak coupling limit the energy spectrum in each Hilbert subspace of an angular momentum consists of discrete LLs of graphene. In the intermediate coupling regime non-resonant states form a closely spaced energy spectrum. We find, counter-intuitively, that resonant quasi-bound states of both positive and negative energies exist in the spectrum. The presence of resonant quasi-bound states of negative energies is a unique property of massless Dirac fermions. As the strong coupling limit is approached resonant and non-resonant states transform into anomalous states, whose probability densities develop a narrow peak inside the well and another broad peak under the potential barrier. These properties may investigated experimentally by measuring optical transition energies that can be described by a scaling function of the coupling constant.
Preface 1. Background in linear algebra 2. Discretization of partial differential equations 3. Sparse matrices 4. Basic iterative methods 5. Projection methods 6. Krylov subspace methods Part I 7. Krylov subspace methods Part II 8. Methods related to the normal equations 9. Preconditioned iterations 10. Preconditioning techniques 11. Parallel implementations 12. Parallel preconditioners 13. Multigrid methods 14. Domain decomposition methods Bibliography Index.
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We calculate the distribution of incompressible and compressible regions at an electrostatically defined edge of a two-dimensional electron system (2DES) in a strong perpendicular magnetic field B for two models, one with a strictly 2D arrangement of gate and charges, and one with a realistic 3D structure. We ensure electrochemical equilibrium by self-consistent calculation of the electrostatic potential from the electron density and the boundary conditions defined by the metallic gates, and of the electron density in this potential, using the Thomas-Fermi approximation. Both models yield qualitatively the same results. In the limit of zero temperature and strong magnetic field, and at distances from the edge which are much larger than the effective Bohr radius, our 2D model yields quantitatively the same result for the density profile in the 2DES as recent work by Chklovskii, Shklovskii, and Glazman, who considered a purely electrostatic model, without requiring electrochemical equilibrium for B=0. We demonstrate explicitly how the incompressible regions are destroyed with increasing temperature.
We describe monocrystalline graphitic films, which are a few atoms thick but are nonetheless stable under ambient conditions,
metallic, and of remarkably high quality. The films are found to be a two-dimensional semimetal with a tiny overlap between
valence and conductance bands, and they exhibit a strong ambipolar electric field effect such that electrons and holes in
concentrations up to 1013 per square centimeter and with room-temperature mobilities of ∼10,000 square centimeters per volt-second can be induced by
applying gate voltage.